Extension operators on balls and on spaces of finite sets

We study extension operators between spaces of continuous functions on the spaces $σₙ(2^{X})$ of subsets of X of cardinality at most n. As an application, we show that if $B_{H}$ is the unit ball of a nonseparable Hilbert space H equipped with the weak topology, then, for any 0 < λ < μ, there is no extension operator $T: C(λB_{H}) → C(μB_{H})$.